2.21. Plot the following curves, locate any fixed points, and obtain asymp- totic expansions to their solution as k y% +a?, a = Yk (yf+3) 3y+1 (a) Yk+1 = real number, (b) Yk+1 = (c) 2y+1 - 5yk+1Yk + 2y% = 2, ayk+1 (d) Yk+1 = Ykta ? 5y-6yk+2 6y -8yk+3' 5y +6yk+19 Y+5 (е) Ук+1 (f) 5yk+1 (g) Yf+1 = Yk + 6.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Solve the question in the same way as a book
![2.8.2
Example B
Consider the following linear difference equation:
3yk+1 = Yk + 2.
(2.177)
This equation has a fixed point yk = 1. The exact solution is given by the
expression
Yk = 1+ A3-k
(2.178)
where A is an arbitrary constant. Consideration of both Figure 2.6 and the
result of equation (2.178) shows that the fixed point is stable.
The linear difference equation
Yk+1 = 2yk – 1
(2.179)
has a fixed point yk = 1 and its exact solution is
Yk = 1+ A2*,
(2.180)
where A is an arbitrary constant. For this case, the fixed point is unstable.
See Figure 2.7.
Likewise, the equation
Yk+1 = -2yk +3
(2.181)
has the fixed point yk = 1. Since the slope is larger in magnitude than one,
the fixed point is unstable. The exact solution is
Yk = 1+ A(-2)*,
(2.182)
where A is an arbitrary constant.
Note that for these three examples, we have, respectively, monotonic con-
vergence, monotonic divergence, and oscillatory divergence. See Figures 2.6,
2.7, and 2.8.
Yk+1
97
(1 + 1)
Yk
FIGURE 2.7: Yk+1 = 2Yk – 1.
Difference Equations
Yk+1
Yk
FIGURE 2.8: Yk+1 = -2yk + 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9396dbf6-1a2a-41f6-9ea8-33e71c92a8b7%2F8ed76f58-6718-4885-a0fe-8b41c33cca82%2F88uusm5_processed.jpeg&w=3840&q=75)
![2.21. Plot the following curves, locate any fixed points, and obtain asymp-
totic expansions to their solution as k
(a) Yk+1 = y + a²,
Yk (Y%+3)
3y%+1
a = real number,
(b) Yk+1 =
(c) 2y+1 - 5yk+1Yk + 2y% = 2,
ayk+1
(d) Yk+1
Yk ta ?
(е) Ук+1
5y% -6yk+2
6y? -8yk+3'
(f) 5yk+1
5y +6yk+19
Y+5
(g) y+1 = Yk +6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9396dbf6-1a2a-41f6-9ea8-33e71c92a8b7%2F8ed76f58-6718-4885-a0fe-8b41c33cca82%2F5qsmgnw_processed.jpeg&w=3840&q=75)
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